Aeta Meehaniea 43, 27--35 (1982) A C T A M E C H A N I C A | by Springer-Verlag 1982

Behaviour of a Discontinuity at the Wave-Head Propagating

Through a Relaxing Gas

By

J. Sharma, R. Shyam, and V. D. Sharma, Varanasi, Ind ia

(Received July 17, 1980; revised February 20, 1981)

Summary

The rate of amplification of a discontinuity in the velocity gradient is evaluated at the wave-head in a spatially uniform but time evolving flow of a relaxing gas. The paper investigates the effects of relaxation (present in the flow) and the initial wave front curva- ture on the growth and decay behaviour of waves in the thermodynamical state of weak or strong equilibrium.

1. Introduction

Propagation of weak waves in ideal gases, which are in a constant state at rest before the arrival of the wave front, has been discussed by Thomas [9], Pack [6], Jeffrey and Taniuti [5] and Prasad and Tagare [7]. In non-equilibrium flows, the propagation of weak waves has been studied by a number of research workers previously (a short account is given by Becker [2] and Clarke and McChesney [4] (p. 142)). In a recent paper Sharma [8] has discussed the growth and decay behaviour of weak discontinuities propagating through a uniform atmosphere of an ideal dissociating diatomic gas; he arrives at the conclusion that in an expanding gas motion, the effects of dissociation reaction are to delay the onset of a shock wave. The purpose of the present paper is to discuss the rate of amplification of a discontinuity at the wave-head propagating through a non-uniform atmosphere of a general relaxing gas; the medium ahead of the wave is assumed to be spatially uniform but of course time dependent. Special attention is paid to the effects of weak or strong equilibrium states. According" to ]3owen [I], a weak equilibrium state is one in which the rate of internal trans- formation is.zero but the affinity of the internal transformation is non-zero; on the other hand a state with both of these quantities zero is defined to be a strong equilibrium state. The influence of a weak equilibrium state on the growth and decay behaviour of waves has not drawn the attention of research workers. We find that a weak equilibrium state may expedite the formation of a shock, no matter how small be the initial discontinuity at the wave-head; this is in contrast to the behaviour of a strong equilibrium state which always delays the formation of a shock at the wavehead (see, [8]).

0001-5970/82/0043/0027/$01.80

28 J. Sharma, t~. Shyam, and V. D. Sharma:

2. Basic Equations

In the absence of viscosity, heat conduction and ,body forces, the equations governing the one dimensional unsteady motion of a relaxing gas are [3] (p. 37)

pt -l- Upx @ O~ep(Ux 4. ]x-~u) == - - c ~ ( A o~8 4. ~q) co(p, s, q), (1)

~ u t @ ~ U U x 4 . P x - - O , (2)

T(st 4- us~) ~- Ao)(p, s, q), (3)

qt 4- Uqx = o)(p, 8, q), (4)

where the letter subscripts denote partial differentiation unless stated otherwise and the constant ] takes values 0, 1, and 2 for plane, cylindrical and spherical waves respectively. The symbols appearing in (1) to (4) are as follows: t is the time, x is the single spatial coordinate being either axial in flows with planar geometry or radial in cylindrically and spherically symmetric flows, u is the gas velocity, p is the pressure, s is the entropy, q is the progress variable eharac- terising the extent of internal transformation in the gas, co is the rate of internal transformation which is assumed to be a known function of p, s and q, and cf is the frozen sound speed given by cf 2 = (~P/8~)s.q. The symbols ~o, T and A, which denote respectively the density, temperature, and the affinity of internal transformation characterized by the variable q, are regarded as functions of p, s and q, and are given by

~-1 ~_ hp, T = h~ and A = --hq,

where h is the specific enthalpy related to p, 8 and q through the canonical equation of state h = h(p, s, q), and the letter subscripts p, s or q denote partial derivatives with respect to the indicated variable while holding the remaining in the set (p, s, q) fixed. I t may be noted that the equation of continuity

et § ue~ 4- euz 4- ] x - l e u ~- 0 (5)

has been used in writing the Eq. (1).

3. Equations in t?haracteristic Form

Let us introduce characteristic parameters ~ and fl which are functions of x and t and which are such that

dx d-t = x~/t~ = u 4. cf where fi(x, t) -= constant (6)

dx - - ~ x~/t~ ~- u - - c~ where c~(x, t) ~- constant (7) dt

where we have written x~ for (~x/Sa)z etc.

Behaviour of ~ Discontinuity at the Wave-Head 29

The t r ans fo rmat ion f rom the space- t ime (x, t) p lane to the plane of the char- acteristic pa rame te r s (~, fi) will be one to one if the Jaeobian .

J • x:t~ - - x~t~ = 2cyt~t ~ (8)

does not vanish anywhere . The f011owing results are noted for fu ture use:

/ ; ~- J - l ( i . t B - - /~ t~) , (9)

h q- u]x := c1J-~(/~t ~ q- ]zt:), {10)

where ] can be either p, u, s or q etc. E q s . (2), (3), (4) and (5) wi th the help of (8), (9) and (10), can be expressed

into the following equivalent form

(19. ~- Qclu~ ) t~ - - (PB - - ~c]u~) t. ---- O, (11)

s~t B + spt. = 2 A ~o(p, s, q) t.t~, (12) T

q.tp + q~t. = 2~o(p, s, q) t.t~, (13)

Also Eqs. (1) and (2) can be expressed as

p: + ecju~ d- c1~{(~a)(P, 6, q) + ]x -~U} t~ = 0, (15)

p~ - - @cyu~ q- c]~{aa)(p, s, q) q- ]xi-leu} t~ = O, (16) Where

A = ~ ~ + ,oq. (17)

4. Evolut ion of Discontinuity at the Wave-Head

The unper tu rbed field ahead of the wave whose behaviour is to be invest igated is assumed to be spat ia l ly uniform. All x derivat ives, therefore, vanish and (1) to (5) yield

Pot :~ --c~0~0~00(P0, so, q0), u0 = 0, Sot = (Ao/To) OJo(Po, So, qo), (18, 19, 20)

qot = c~ so, qo), Qo ~- cons tan t , (21, 22)

where the subscr ipt - - 0 indicates a value in the region ahead of the wave front fl(x, t ) ~ O, or x - ~ x(t) where x(t) denotes the posit ion on the wave front a t any t ime t. The per tu rba t ions of the s ta te ahead are assumed to p ropaga te through the relaxing gas behind the wave f ront fi = 0. Cont inui ty of the variables p, u, o, q and ~ a t fl ~- 0, is essential bu t discontinuit ies in their der ivat ives are per- mi t ted . Any der ivat ive with respect to ~ mus t be cont inuous; discontinuities can appea r only in the fl derivat ives. The associated discontinuit ies in the x der ivat ives follow f rom (9).

Different ia t ion of (15) with respect to fi and of (16) wi th respect to ~ followed

30 J. Sharma, 1~. Shyam, and V. D. Sharma:

by subt rac t ion and evaluat ion a t fl = 0 -f- gives

2~oCloU~+~ + (~OoClo)~ u f + ]x-l~oc~oto~u~ + + 2Cjo(~o~Oo(C~to. Clouts+ ) (23)

+ C~oao(wp+to~ - - wo:t~) + C]oWo(afto: - - ao,t~ +) O,

where the quant i t ies wi th a subscr ipt - - 0 are continuous across fl = 0, while those wi th a superscr ipt + m a y be discontinuous a t the wave head, so t ha t their values just behind the f ront are required in (23).

Eva lua t ing (11), (12), (13) and (14) a t fl = 0 + and using (18) to (22), we get

F r o m the forms of cj,

p f to: - - po~t~ + = O~ oCloUp+ to~ ,

S f to~ - - Soft f : O,

q f to~ - - qo~t f = 0,

CfoO~fl § : O oU~ + .

og(p, s, q) and a it t ranspires t ha t

c~to~ - - Clouts§ : O~o( ~Ci2/ ~p)o u~+toJ2,

~o f to~ - - wo.t~ + = eoC]o( ~Og/ ~p) 0 u f to.,

cr~+to~ - - ao.O + = O~oC]o(~(r/~p)o u~+to~,

where use is made of (24), (25) and (26). Eq. (23), with the help of (28), (29) and (30), yields

where 1

A 1 : -~- {C~oO'o(Cq~/~p)o + 090(70(~Cf2/~0)0 ~- C~O0)O(5q,O'/G~_~D)O } .

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

Eq. (31) gives the var ia t ion of u S with ~, which on in tegra t ion between ai (where u~ + = u~: i and t = ti) and ~ yields

/:// } = exp A , + (32) Ufl + \ eoC]o / ~ 2x ]

The veloci ty gradient at the wave head can be expressed as

Ux + : - -u~+/2c /o t f , (33) which yields

u s = - -uA2c]oi t~ , , (34)

where use is made of (8), (9) and (19). Eqs. (6) and (7) combine to give

2cfl.~ + (u~ + ci~ ) t. - - (u. - - cI~ ) t~ = O, (35)

so tha t a t f i = 0 +

2Clo(t~+): + u f t o . + {c~to: + cjo~t~ +} = O. (36)

Behaviour of a Discontinuity at the Wave-Head 31

I t follows f rom cs2 -- (@/~o~)s. q t h a t

2c/(c/e4 + c/.te) = (ac/2/~p) (pzt. + p.te)

+ (ac/=/as) % 4 + s~t~) + (&:Vaq) (qpt~ + q~tp). (3v)

Combining (15) and (16), we find, on eva lua t ing a t t3 = 0 ~- and using (18) and (19), t ha t

pSto~ ~- po~t~ + ~ {QoC:oUp + -- 2C~oUo~Oot~+} to.. (38)

Eva lua t ing (37) a t fi : 0 -}- and using (25), (26) and (38), we get

c~to~ + c:o,tff = { ~ - ~ u~+ -- (&/2/~p)o c:o%a)ot~+} to~ , (39)

where Co = 1 q- ~o(ac/2/@)o is a t he rmodynamic quan t i t y which is a funct ion of Po, so and qo. For an ideal gas, however, it m a y be readi ly verified t ha t Fo ~- y, the adiabat ic index. Then (36) wi th (39) becomes

where

(tp+). -- A2to=tff -}- i (Fo § i_______~) us+to. = O, (40) 4 Cfo

1 & = T (~cf2/@)~ ~oOo.

Eq. (40), together with the result (32), can be in tegra ted to f ind the var ia t ion of t~ + with t ime a t the wave head . In tegra t ion of (40) yields

( / ) [ / {:/I 1,] 4Clo \ ~ /

(41) where

1 ~o = & + & - T {2~~176176 + ~:~o(aO,/@)o + ~oC~o(a~/@)o} (42)

and u~i is the value of us + when t ~ t i .

Eq. (33), toge ther with (32), (34) and (41), yields

Ux~ (eoiC/ol/eoC]o) exp iz o + dt

u , + : ~ , (43)

t~ ( h

which gives the var ia t ion of the discont inui ty in u , + a t fl :~ 0. I n any event it is clear f rom (43) t ha t the t empora l behaviour of the veloci ty gradient a t the wave head will depend crit ically on the sign of/z 0.

32 J. Sharma, P~. Shyam; and V: D. Sharma:

5. Discussion

Caee (I). When the medium ahead is in a uniform state with respect to space and time both, the quantities Pc, So, qo and Czo are constants, and hence x ----- x o + cfot; xo being the position of the wave front at time t ~- 0 which is either positive or negative, respectively, according as the wave is diverging or converging. Here ti has been set equal to zero for convenience. Thus, Eq. (43) reduces to

Ux + = u~i(x/xo)-J~2 exp (--#o t) (44)

/ ( f o + 1) U+x~ (X/Xo)-il2 exp (--~0 f) df i + 2 0

I t is clear from (44) that in this case also the behaviour of %+ depends critically on the sign of consfant #0. According to Bowen [1], we find that in a strong equilibrium state (COo ~- A0 = 0), Eq. (42), i n view of (17), becomes

2 J 1) ~o = T (oq)o Cio(a~ol@)o = ~o i4o

where ~o = --(aoJ/Sq)o -1 is a relaxation time of the medium and Cso is the equi- 2 librium speed of sound defined as Cr =- (@/@)~, q ~- q~. Since Cjo > %o, it follows

that in a strong equilibrium state #o > 0. In a weak equilibrium state (co o = 0, and Ao # 0), Eq. (42) becomes

l c 2 {(~q)o- (~ )o @~)o} ( & d @ ) o ;

which shows, that in a weak equilibrium state/z 0 may be either positive or negative or zero. The case /~o = 0 corresponds exactly with that discussed by Thomas [9] and Pack [6] and the ease ,u0 > 0 has been discussed in [8]. We thus consider the case when ~o < 0.

(i) Diverging Waves (Xo > 0): When #o < 0 (which can occur only in a weak

equilibrium state) then the integral / (X/Xo) -~12 exp (--#ot) dt diverges to d-co and 0

hence the initial critical value of the discontinuity vanishes. Thus, Eq. (44) whows that, for #o < 0, all expansive Waves decay and ultimately take a stable wave form, i.e. for %~ > 0, uz + --> 2 [/~0i (F0 + 1) -1 as t -+ 0% while all compressive waves always steepen up into shock in a finite time, no matter how small be

+ their initial discontinuity, i.e. when u,~ ~ 0, luff[--> e~ as t-->~s, where tc is given by

2 I/~ol [ for ] = 0 ~ = (1/1~,ol) log I + (G T- ~ >klJ '

and

j (X/Xo) -~/2 exp (t#ol t) dt = 2{(/'o + 1) lux~l} -~ for j = 1, 2. 0

Behaviour of a Discontinuity at the Wave-Head 33

I n this case, (~c/~ I/~01)< 0 and (~'4/~Xo)< 0 which mean tha t the weak equilibrium state (with /~0 < 0) causes a compression wave to steepen into a shock more swiftly than it does in an inert a tmosphere (in which/~o = 0) whereas an increase in the initial wave f ront curvature (1/xo) delays the shock formation.

(ii) Converging waves (x0 < 0) : Here we consider the following two cases:

Cylindrical waves: I n this case Eq. (44) reduces to

u~{1 + (%t/Xo)}-~; ~ e~p ( - z00 (45) ~ x + ~

1 + (F~ + 1_____~) u+ii(t ) 2

where t

I(t) = f {1 + (C, fj/XO)} -1[2 ex]~ (--~0 ~) dr. (46) 0

In this situation, irrespective of the sign of ~t0, the numera tor of (45) becomes unbounded at t* ~ (Ixol/C/o) whereas the denominator , for ux+/> 0, remains positive and finite since the integral I(t) converges to a finite positive l imit as

+ t--~ t*. Hence, Ux+--> ~ as t - + t* when uxi > 0. This corresponds to the for- + mat ion of a focus but not to the shock. I f uxl < 0 then there exists a positive

critical value (Ux+)c+ given by

[ ; (ux+)~§ ___ (To + 1) {1 + (clot/Xo)} -~/2 exp (--/~0t) dt 2

o

such tha t for Iux+it < (ux+)c§ [ux+l --> eo as t --> ~ (i.e. a focus occurs bu t not the shock) and for lUx~l > (u~+)c~, lUx +] --+ ~ as t --> 4 + where 4 + < t* is the unique solution of the equat ion

tr + + --1

0

Since 4 + < t* it indicates t ha t the shock forms before the format ion of the focus. I f [us I ~ (u~+)~+, it is obvious f rom (45) tha t lux+l -+ oo as t --> t*, i.e. the shock and focus are formed together. Fur ther , since (04+/~ Ix01) > 0, it follows tha t an increase in the initial wave front curvature (l/x0) reduces the shock format ion time.

I t is interesting to note that , like a s t rong equilibrium (see, [8]), a weak equilibrium state with #0 < 0 also possesses the proper ty tha t a cylindrically converging compressive wave terminates into a shock only if the magni tude of the initial discontinui ty exceeds a critical value, this is in contrast to the cor- responding case of a diverging cylindrical wave in a state of weak equilibrium state with #0 < O, discussed above.

Spherical waves: I n this case Eq. (44) takes the form

u~+~{1 + (eot/Xo)} -~ exp (-~o0, (47) U X + ~

+ (to + ~_____~) u+j(t) 2

3 Acta M e c h . 43/1--2

34 J. Sharma, R. Shyam, and V. D. Sharma:

where t

I(t) = f {1 + (cfjIxo)} -~ exp (-~j) 4~. 0

(48)

In this case, i rrespective of t h e sign of #0, bo th the numera to r of (47) and the integral given b y (48) become unbounded at a finite t ime t * * = (IXol/elo). Hence, it follows f rom (47) t ha t for Uxi > o, lux+l :~ ~ as t ~ t**, i.e. a focus is formed. But if Uxi < 0 then there exists a finite t ime tc ++ < t** such t h a t lUx +1 ---> oc as t -~ t~ ++, where t~ ++ is given b y

te ++

f {1 - - (Clot/lXol)} -1 exp ( - -#ot)dt = {(/'o + 21)]u~il.} -1 " 0

In this si tuation, we, thus, arr ive at an interest ing result that, in bo th the equil ibrium states, all spherical ly converging compress ive waves, no m a t t e r how small be their initial discontinuity, a lways end u p into shock waves before the fo rmat ion of th e focus; this is in contras t to the corresponding ease of cylin- dr ical ly converging waves.

Case ( I I ) . I f the med ium ahead is non-uni form i.e. the med ium ahead of the wave is spat ia l ly uni form bu t t ime evolving with #0 < 0 a n d one considers only a short in terval of t ime, so t ha t the quant i t ies Cg0, F0 and #0 do not change ap- preciably between ti and t, it is evident tha t (43) can be wri t ten in the approxi- ma te fo rm

u+~{Xo/(Xo -1- C, fot)}J 12 e x p (--riot) ux+ --~ t (49)

1 + (FO+ ~) u ~ f 2 {x~176 + cf~ exp (--rio ~') d~" , 2

0

where ~1o, /~o and Fo indicate suitable mean values over the interval ti to t, and ti has been set equal to zero for convenience.

An examina t ion of (49) leads to the conclusion t ha t if Ux~,+ tic are negative. and P0, Xo are posi t ive, then there exists a finite t ime t s given b y

2 Ifi0] ~ for ] - - 0 (50) t~ = (1/I/7o[) log 1 + (/'~o + 1)fu2dJ'

and t.

= luxil} for ] 1, 2 (5l) f {Xo/(Xo + ~fot)}i/2 exp ([fi0[ t) dt 2{(_F 0 + 1) + -~ = 0

such tha t lUx+[ -> cx~ as t --> t~ provided (49) remains valid over the requisite period. Thus, we find t ha t in a non-uni form a tmosphere (with rio < 0), a dis- cont inui ty associated w i t h ~ compression wave, no m a t t e r how small, a lways steepens up into a shock af ter a finite t ime. I t is evident f rom (50) and (51) tha t (~t~/~ ]f i0])< 0 and ( ~ t J ~ x o ) < 0 which m e a n tha t the non-unif0rm at- mosphere (with rio < 0) accelerates the steepening tendency of a compression wave to grow into shock whereas an increase i n 6urvature causes the shock fo rmat ion t ime to increases.

Behaviour of a Discontinuity at the Wave-Head 35

References

[1] Bowen, R. M. : Acceleration and higher order waves is a mixture of chemically reacting elastic materials. Arch. l~atl. Mech. Anal, 38, 169--180 (1969).

[2] Becker, E.: Chemically reacting flows. Annual tCeview of Fluid Mech. 4, 168--175 (1972).

[3] Chu, B. T. : In non equilibrium flows, part I I (Wegener, P. P., ed.). New York: Marcel Dekker 1970.

[4] Clarke, J. F., McChesney, M. : Dynamics of relaxing gases. London: Butterworths 1976.

[5] Jeffrey, A., Taniuti, T. : Non-linear wave propagation. New York: Academic Press 1964.

[6] Pack, D. C. : A note on the breakdown of continuity in the motion of a compressible Iluid. J . Fluid Mech. 8, 103--108 (1960).

[7] Frasad, P., Tagare, S. G. : On the breakdown of the continuity in the motion of a compressible fluid with variable density in an ambient medium. ZAMP 22, 359--364 (1971).

[8] Sharma, V. D. : Weak discontinuities in a dissociating gas. Acta Mech. 88, 249--256 (1981).

[9] Thomas, T. Y. : Tim growth and decay of sonic discontinuities in ideal gases. J. Math. IY[eeh. 6, 455--469 (1957).

J. Sharma, R. Shyam, and V. D. Sharma* School o/ Applied Sciences

Institute o/Technology Banaras Hindu University

Varanasi 221005, India

* P~esent address: Department o] Aerospace Engineering

University o/ Maryland College Park, MD 207d2, U.S.A.

3*